Privacy Preserving Statistical Analysis for Distributed Databases

Abstract

Aggregate statistics are determined by first randomizing independently data X and Y to obtain randomized data {circumflex over (X)} and . The first randomizing preserves the privacy of the data X and Y. Then, the randomized data {circumflex over (X)} and is randomized secondly to obtain randomized data {tilde over (X)} and {tilde over (Y)} for a server, and helper information T.sub.{tilde over (X)}|{circumflex over (X)} and T.sub. | for a client, wherein T represents an empirical distribution, and wherein the randomizing secondly preserves the privacy of the aggregate statistics of the data X and Y. The server then determines T.sub.{tilde over (X)},{tilde over (Y)}. Last, the client applies the side information T.sub.{tilde over (X)}|{circumflex over (X)} and T.sub. | to T.sub.{tilde over (X)},{tilde over (Y)} to obtain an estimated {dot over (T)}_X,Y, where "|" and "," between X and Y represent a conditional and joint distribution, respectively.

Description

FIELD OF THE INVENTION

[0001] This invention relates generally to secure computing by third parties, and more particularly to performing secure statistical analysis on a private distributed database.

BACKGROUND OF THE INVENTION

[0002] Big Data

[0003] It is estimated that 2.5 quintillion (10¹⁸) bytes of data are created each day. This means that 90% of all the data in the world today has been created in the last two years. This "big" data come from everywhere, social media, pictures and videos, financial transactions, telephones, governments, medical, academic, and financial institutions, and private companies. Needless to say the data are highly distributed in what has become known as the "cloud,"

[0004] There is a need to statistically analyze this data. For many applications, the data are private and require the analysis to be secure. As used herein, secure means that privacy of the data is preserved, such as the identity of the sources for the data, and the detailed content of the raw data. Randomized response is one prior art way to do this. Random response does not unambiguously reveal the response of a particular respondent, but aggregate statistical measures, such as the mean or variance, can still be determined.

[0005] Differential privacy (DP) is another way to preserve privacy by using a randomizing function, such as Laplacian noise. Informally, differential privacy means that the result: of a function determined on a database of respondents is almost insensitive to the presence or absence of a particular respondent. Formally, if the function is evaluated on adjacent databases differing in only one respondent, then the probability of outputting the same result is almost unchanged.

[0006] Conventional mechanisms for privacy, such as k-anonymization are not differentially private, because an adversary can link an arbitrary amount of helper (side) information to the anonymized data to defeat the anonymization.

[0007] Other mechanisms used to provide differential privacy typically involve output perturbation, e.g., noise is added to a function of the data. Nevertheless, it can be shown that the randomized response mechanism, where noise is added to the data itself, provides DP.

[0008] Unfortunately, while DP provides a rigorous and worst-case characterization for the privacy of the respondents, it is not enough to formulate privacy of an empirical probability distribution or "type" of the data. In particular, if an adversary has accessed anonymized adjacent databases, then the DP mechanism ensures that the adversary cannot de-anonymize any respondent. However, by construction, possessing an anonymized database reveals the distribution of the data.

[0009] Therefore, there is a need to preserve privacy of the respondents, while also protecting an empirical probability distribution from adversaries.

[0010] In U.S. application Ser. No. 13/032,521, Applicants disclose a method for processing data by an untrusted third party server. The server can determine aggregate statistics on the data, and a client: can retrieve the outsourced data exactly. In the process, individual entries in the database are not revealed to the server because the data are encoded. The method uses a combination of error correcting codes, and a randomization response, which enables responses to be sensitive while maintaining confidentiality of the responses.

[0011] In U.S. application Ser. No. 13/032,552. Applicants disclose a method for processing data securely by an untrusted third party. The method uses a cryptographically secure pseudorandom number generator that enables client data to be outsourced to an untrusted server to produce results. The results can include exact aggregate statistics on the data, and an audit report on the data. In both cases, the server processes modified data to produce exact results, while the underlying data and results are not revealed to the server.

SUMMARY OF THE INVENTION

[0012] The embodiments of the invention provide a method for statistically analyzing data while preserving privacy of the data.

[0013] For example, Alice and Bob are mutually untrusting sources of separate databases containing information related to respondents. It is desired to sanitize and publish the data to enable accurate statistical analysis of the data by an authorized entity, while retaining the privacy of the respondents in the databases. Furthermore, an adversary must not be able to analyze the data.

[0014] The embodiments provide a theoretical formulation of privacy and utility for problems of this type. Privacy of the individual respondents is formulated using ε-differential privacy. Privacy of the statistics on the distributed databases is formulated using δ-distributional and ε differential privacy.

[0015] Specifically, aggregate statistics are determined by first randomizing independently data X and Y to obtain randomized data {circumflex over (X)} and . The first randomizing preserves a privacy of the data X and Y.

[0016] Then, the randomized data {circumflex over (X)} and is randomized secondsly to obtain randomized data {tilde over (X)} and {tilde over (Y)} for a server, and helper information on T.sub.{tilde over (X)}|{circumflex over (X)} and T.sub. | for a client, wherein T represents an empirical distribution, and wherein the randomizing secondly preserves the privacy of the aggregate statistics of the data X and Y.

[0017] The server then determines T.sub.{tilde over (X)},{tilde over (Y)}. Last, the client applies the the side information T.sub.{tilde over (X)}|{circumflex over (X)} and T.sub. | to T.sub.{tilde over (X)},{tilde over (Y)} obtain an estimated {dot over (T)}_X,Y, wherein "|" and "," between X and Y represent a conditional and joint distribution, respectively.

BRIEF DESCRIPTION OF THE DRAWING

[0018] FIG. 1 is a flow diagram of a method for securely determining statistics on private data according to embodiments of the invention;

[0019] FIG. 2 is a block diagram of private data from two sources operated on according to embodiments of the invention;

[0020] FIG. 3 is a schematic of a method according to embodiments of the invention for deriving statistics from the data of FIG. 2 by a third party without compromising privacy of the data; and

[0021] FIG. 4 is a schematic of an application of the method according to embodiments of the invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0022] Method Overview

[0023] As shown in FIG. 1, the embodiments of our invention provide a method for securely performing statistical analysis on private data. This means the actual raw data is not revealed to anyone, other than sources of the data.

[0024] In security, privacy and randomization applications "weak" and strong" are terms of art that are well understood and documented. Weak means that underlying data (e.g., password, user identification, etc.) is could be recovered with know "cracking" methods. Strong means that the data is very difficult to recover in given a reasonable amount of time and reasonable computing resources.

[0025] In addition, the randomization means randomizing the data according to a particular distribution. The term encompasses the following concept. First, the data are anonymized to protect privacy. Second, data are sanitized to reinforce the notion that the operation serves the purpose of making the data safe for release.

[0026] Data X 101 and Y 102 are first randomized (RAMI) independently to obtain randomized data {circumflex over (X)} and , respectively. The randomizations 110 and 115 can be the same or different. In the preferred embodiment, we use a Post RAndomisation Method (PRAM). The security provided by 110 and 115 is relatively "weak." This means that the identities of data sources are hidden and individual data privacy is preserved, but aggregate statistics on the data could perhaps be determined with some effort.

[0027] The randomized data {circumflex over (X)} and data are again (second) randomized (RAM2) to obtain randomized data {tilde over (X)} and {tilde over (Y)} for a server, and helper information T.sub.{tilde over (X)}| and T.sub. | for a client, respectively. The second randomizations can be the same or different than the first randomizations. In the helper information, T represents a true empirical distribution.

[0028] In statistics, an empirical distribution is the normalized histogram of the data. Each of n data points contributes by 1/n to the empirical distribution. The empirical distribution is representative of the underlying data. The emperical distribution is sufficient to determine a large number of different types of statistics, including mean, median, mode, skewedness, quantiles, and the like.

[0029] The security provided by 120 and 125 is relatively "strong." That is, the privacy of aggregate statistics on the data X and Y is preserved.

[0030] The server 130 determines T.sub.{tilde over (X)},{tilde over (Y)}{tilde over ( )} after {tilde over (X)} and {tilde over (Y)}0 are combined.

[0031] The client can now apply the side information T.sub.{tilde over (X)}|{circumflex over (X)} and T.sub. | to T.sub.{tilde over (X)},{tilde over (Y)} to "undo" the second randomization, and obtain an estimated {dot over (T)}_X,Y. The estimated, indicated by above, distribution of the data X and Y is sufficient to obtain first, second, etc. order statistics. Although the client can determine statistics, the client cannot recover the exact data X and and Y because of the weak security.

[0032] Method Details

[0033] For ease of this description as shown in FIG. 2, we present our problem formulation and results with two data sources Alice and Bob. However, our method can easily be generalized to more than two sources. Also, other levels of security with fewer or more randomizations can also be used.

[0034] Alice and Bob independently sanitize 210 data 201-202 to protect the privacy of respondents 205. As used herein, it is not possible to recover exact private information from sanitized data. A number of techniques are know for sanitizing data, e.g., adding random noise.

[0035] The sanitized data 211-212 are combined 220 into a database 230 at a "cloud" server. The server can be connected to a public network (Internet). This is the data is available for statistical analysis by an authorized user of a client.

[0036] As shown in FIG. 3, Alice and Bob store the sanitized data in at the server to facilitate transmission and computation required on these potentially large databases. An entrusted authorized client 301 can now perform statistical analysts on the data with the assistance of low-rate helper-information 303. The helper information is low-rate in that it is relatively small in comparison to the original database and/or the randomized data. The helper information 303 allows the authorized client to essentially undo the second randomization.

[0037] The analysis is subject to the following requirements. The private data of the sources should not be revealed to the server or the client. The statistics of the data provided by sources and Bob should not be revealed to the server. The client should be able to determine joint, marginal and conditional distributions of the data provided by Alice and Bob. The distributions are sufficient to determine first, second, etc. order statistics of the data.

[0038] Problem Framework and Notation

[0039] The Alice data are a sequence of random variables X:=(X₁,X₂, . . . , X_n), with each variable X_i taking values from a finite-alphabet X. Likewise, Bob's data are modeled as a sequence of random variables Y:=(Y₁,Y₂, . . . , Y_n), with each Y_i taking values from the finite-alphabet Y. The length of the sequences, n, represents the total number of respondents in the database, and each (X_i,Y_i) pair represents the data of the respondent i collectively held by Alice and Bob, with the alphabet X×Y representing the domain of each respondent's data.

[0040] data pairs (X_i,Y_i) are independently and identically distributed (i.i.d.) according to a joint distribution P_X,Y over X×Y, such that for

x := ( x 1 , , x n ) .di-elect cons. X n , and ##EQU00001## y := ( y 1 , , y n ) .di-elect cons. Y n , such that P X , Y ( x , y ) = i = 1 n P X , Y ( x i , y i ) . ##EQU00001.2##

[0041] A privacy mechanism randomly maps 310 input to output, M: I→O, according to a conditional distribution P_O|I. A post RAndomisation method (PRAM) is a class of privacy mechanisms where the input and output are both sequences. i.e., I=O=Dⁿ for an alphabet D, and each element of the input sequence is i.i.d. according to an element-wise conditional distribution.

[0042] Alice and bob each independently apply PRAM to their data as R_A:Xⁿ→Xⁿ and R_B:Yⁿ→Yⁿ. The respective outputs are

{tilde over (X)}:=({tilde over (X)}₁, . . . , {tilde over (X)}_n):=R_A(X)

and

{tilde over (Y)}:=({tilde over (Y)}₁, . . . , {tilde over (Y)}_n):=R_B(Y),

and the governing distributions are

[0043] P.sub.{tilde over (X)}|X and P.sub.{tilde over (Y)}|Y,

so we have that

P X ~ , Y ~ X , Y ( x ~ , y ~ x , y ) = P X ~ X ( x ~ x ) P Y ~ Y ( y ~ y ) = i = 1 n P X ~ X ( x ~ i x i ) P Y ~ Y ( y ~ i y i ) ##EQU00002##

[0044] We also use R_AB: Xⁿ×Yⁿ→Xⁿ×Yⁿ, defined by

R_AB(X, Y):=({tilde over (X)}, {tilde over (Y)}):=(R(X), R_B(Y))

to denote a mechanism that arises from a concatenation of each individual mechanism. R_AB is also a PRAM mechanism and is governed by the conditional distribution P.sub.{tilde over (X)}|XP.sub.{tilde over (Y)}|Y.

[0045] Type Notation

[0046] The type or empirical distribution of the sequence of the random variables X=(X₁, . . . , X_n) is the mapping T_X:X→[0,1] defined by

T X ( x ) := { i : X i = x } n , .A-inverted. x .di-elect cons. X . ##EQU00003##

[0047] A joint type of two sequences X=(X₁, . . . , X_n) and Y=(Y₁, . . . , Y_n) is the mapping T_X,Y:X×Y→[0,1] defined by

T X , Y ( x , y ) := { i : ( X i , Y i ) = ( x , y ) } n , .A-inverted. ( x , y ) .di-elect cons. X × Y . ##EQU00004##

[0048] A conditional type of a sequence Y=(Y₁, . . . , Y_n) given another X=(X₁, . . . , X_n) is the mapping T_Y|X:Y×X→[0,1] defined by

T Y X ( y x ) := T Y , X ( y , x ) T X ( x ) = { i : ( Y i , X i ) = ( y , x ) } { i : X i = x } ##EQU00005##

[0049] The conditional distribution is the joint distribution divided by the marginal distribution.

[0050] Values of these type mappings are determined, given the underlying sequences, and are random when the sequences are random.

[0051] Matrix Notation for Distributions and Types

[0052] The various distributions, and types of finite-alphabet random variables can be represented as vectors or matrices. By fixing a consistent ordering on their finite domains, these mappings can be vectors or matrices indexed by their domains. The distribution P_X:X→[0,1] can be written as an |X|×1 column-vector P_X, whose x^th element, for x .di-elect cons. X, is given by P_X[x]:=P_X(x).

[0053] A conditional distribution P_Y|X:Y×Y→[0,1] can be written as a |Y|×|X| matrix P_Y|X, defined by P_Y|X[y,x]:=P_Y|X(y|x). A joint distribution P_X,Y:X×Y→[0,1] can be written as a |X|×|Y| matrix P_X,Y, defined by P_X,Y[x,y]:=P_X,Y(x,y), or as a |X∥Y|×1 column-vector P_X,Y, formed by stacking the columns of P_X,Y.

[0054] We can similarly develop the matrix notation for types, with T_X, T_Y|X, T_X,Y and T_X,Y similarly defined for sequences X and Y with respect to the corresponding type mappings. These type vectors or matrices are random quantities.

[0055] Privacy and Utility Conditions

[0056] We now formulate the privacy and utility requirements for this problem of computing statistics on independently sanitized data. According to the privacy requirements described above, the formulation consider privacy of the respondents, privacy of the distribution, and finally the utility for the client.

[0057] Privacy of the Respondents

[0058] The data related to a respondent must be kept private from all other parties, including any authorized, and perhaps untrusted clients. We formalize this notion using ε-differential privacy for the respondents.

[0059] Definition: For ε≧0, a randomized mechanism M:Dⁿ→O gives ε-differential privacy if for all data, sets d,d'.di-elect cons.Dⁿ, within Hamming distance d_H(d,d')≦1, and all S.di-elect cons.O,

Pr[M(d).di-elect cons.S]≦e.sup.εPr[M(d').di-elect cons.S].

[0060] Under the assumption, that the respondents are sampled i.i.d., a privacy mechanism that satisfies DP results in a strong privacy guarantee. Adversaries with knowledge of all respondents except one, cannot discover the data of the sole missing respondent. This notion of privacy is rigorous and widely accepted, and satisfies privacy axioms.

[0061] Privacy of the Distribution

[0062] Alice and Bob do not want to reveal the statistics of the data to adversaries, or to the server. Hence, the sources and server must ensure that the empirical distribution, i.e., the marginal and joint types cannot be recovered from {tilde over (X)} and {tilde over (Y)}. As described above, ε-DP cannot be used to characterize privacy in this case. To formulate a privacy notion for the empirical probability distribution, we extend ε-differential privacy as follows.

[0063] Definition: (δ-distributional ε-differential privacy) Let (`,`) be a distance metric on the space of distributions. For ε,δ≧0, a randomized mechanism M:Dⁿ→O gives δ-distributional ε-differential privacy if for all data sets d,d'.di-elect cons.Dⁿ, with d(T_d, T_d')≦δ, and all S.OR right.O,

Pr[M(d).di-elect cons.S]≦e.sup.εPr[M(d').di-elect cons.S].

[0064] A larger δ and smaller ε provides better protection of the distribution. Our definition also satisfies privacy axioms.

[0065] Utility for Authorized Clients

[0066] The authorized client extracts statistics from the randomized database 230. We model this problem as the reconstruction of the joint and marginal type functions T_X,Y(x,y), T_X(x), and T_Y(y), or (equivalently) the matrices T_X,Y, T_X and T_Y. The server facilitates this reconstruction by providing computation

[0067] based on the sanitized data ({tilde over (X)}, {tilde over (Y)}). Alice and Bob provide low-rate, independently generated helper-information 203. With the server's computation and the helper-information, the client produces the estimates {dot over (T)}_X,Y, {dot over (T)}_X, and {dot over (T)}_Y.

[0068] For a distance metric d(`,`) over the space of distributions, we define the expected utility of the estimates as

μT_X,Y:=E[-d({dot over (T)}_X,Y,T_X,Y)],

μT_X:=E[-d({dot over (T)}_X,T_X)], and

μT_Y:=E[-d({dot over (T)}_Y,T_Y)].

[0069] Analysis of Privacy Requirements

[0070] The privacy protection of the marginal types of the database implies privacy protection for the joint type because the distance function d satisfies a general property shared by common distribution distance measures.

[0071] Lemma 1: Let d(`,`) be a distance function such that

d(T_X,Y,T_X',Y')≧max(d(T_X,T_X'),d(T_Y,T_Y') ). (1)

[0072] Let M_AB be the privacy mechanism defined by M_AB(X,Y):=(M_A(X), M_B(Y)). If M_A satisfies δ-distributional ε₁-differential privacy and M_B satisfies δ-distributional ε₂-differential privacy, then M_AB satisfies δ-distributional (ε₁+ε₂)-differential privacy.

[0073] If vertically partitioned data are sanitized independently and we want to recover joint distribution from the sanitized table, the choice of privacy mechanisms is restricted to the class of PRAM procedures. We analyze the constraints that should be placed on the PRAM algorithms so that they satisfy the privacy constraints. First, consider the privacy requirement of the respondents in Alice and Bob's databases.

[0074] Lemma 2: Let R: Xⁿ→Xⁿ be a PRAM mechanism governed by conditional distribution P.sub.{tilde over (X)}|X. R satisfies ε-DP if

ε = max x 1 , x 2 , x ~ .di-elect cons. X ln ( P X ~ X ( x ~ x 1 ) ) - ln ( P X ~ X ( x ~ x 2 ) ) . ( 2 ) ##EQU00006##

[0075] Lemma 3: Define M_AB(x,y)=(M_A(x), M_B(y)). If M_A satisfies ε₁-DP and M_B satisfies ε₂-DP, the M_AB satisfies (ε₁+ε₂)-DP.

[0076] The lemma can be extended to k sources where if i^th source's sanitized data, satisfies ε_i-DP, then the joint system provides (Σ_i=1^kε_i)-DP. Next, we consider the privacy requirement for the joint and marginal types.

[0077] Lemma 4: Let d(`,`) be the distance metric on the space of distributions. Let R: Xⁿ→Xⁿ be a PRAM mechanism governed by conditional distribution P.sub.{tilde over (X)}|X.

[0078] Necessary Condition: If R satisfies δ-distributional ε-DP, then R must satisfy

n / 2 - DP ##EQU00007##

for the respondents.

[0079] Sufficient Condition: If R satisfies

n - DP ##EQU00008##

for the respondents, then R satisfies δ-distributional ε-DP.

[0080] Example Implementation

[0081] We now describe an example realization of the system framework given above, where the privacy mechanisms are selected to satisfy our privacy and utility requirements. The key requirements of this system can be summarized as follows:

[0082] (I). R_AB is a δ-distributional ε-differentially private mechanism;

[0083] (II). Helper information is generated by a ε-DP algorithm; and

[0084] (III). R_A and R_B are PRAM mechanisms.

[0085] Because the santized data are generated by a δ-distributional ε-differentially private mechanism, helper information is necessary to accurately estimate the marginal and joint type. To generate outputs that preserve different levels of privacy, the sources use a multilevel privacy approach.

[0086] As shown in FIG. 4, the databases are sanitized by a two-pass randomization process 410, see FIG. 1. The first pass R_AB,1 takes the raw source data X,Y as input and guarantees the respondent privacy, while the second pass R_AB,2 takes the sanitized output {circumflex over (X)}, ) of the first pass as input and guarantees distributional privacy. The helper information 303 is extracted during the second pass to preserve respondent privacy. The mechanisms are constructed with the following constraints:

R A , 2 and R B , 2 are 2 n - DP ; ( i ) . R A , 1 and R B , 1 are 2 - DP ; and ( ii ) . ##EQU00009##

[0087] (iii), R_A,1, R_A,2, R_B,1 and R_B,2 are PRAM, mechanisms.

[0088] By Lemma 3, constraint (ii) implies R_AB,1 is ε-DP and hence implies requirement (II). Note that R_A(X) can be viewed as R_A,2(R_A,1) (X)) and is governed by the conditional distribution (in matrix notation)

P.sub.{tilde over (X)}|X=P.sub.{tilde over (X)}|{circumflex over (X)}P.sub.{tilde over (X)}|X.

[0089] Hence, constraint (iii) implies that requirement (III) is satisfied. By Lemmas 1 and 4, constraint (i) implies that requirement (i) is satisfied. Now, all the privacy requirement are satisfied. In the following, we describe how the client can determine the estimated types.

[0090] Recall that without the helper information, the client cannot accurately estimate exact types due to requirement (I). In this example, the helper information includes the conditional types T.sub.{circumflex over (X)}|{circumflex over (X)} and T.sub. | determined during the second pass. An unbiased estimate of T_X determined from {tilde over (X)} is given by P.sub.{tilde over (X)}|X^-1T.sub.{tilde over (X)} and the exact types can be recovered by T.sub.{tilde over (X)}|X^-1T.sub.{tilde over (X)}. Thus, we have the following identities and estimators:

T.sub.{circumflex over (X)}=T.sub.{tilde over (X)}|{circumflex over (X)}^-1T.sub.{tilde over (X)}, (4)

{dot over (T)}_X=P.sub.{tilde over (X)}|{circumflex over (X)}^-1T.sub.{circumflex over (X)}=P.sub.{tilde over (X)}|{circumflex over (X)}^-1T.sub.{tilde over (X)}|{circumflex over (X)}^-1T.sub.{tilde over (x)},

T.sub. =T.sub.{tilde over (Y)}| ^-1T.sub.{tilde over (Y)}, (5)

{dot over (T)}_Y=P.sub.{tilde over (Y)}| ^-1T.sub. =P.sub.{tilde over (Y)}| ^-1T.sub.{tilde over (Y)}| ^-1T.sub.{tilde over (Y)},

[0091] Extending the results to determine the joint type presents some challenges. The matrix form of the conditional distribution of the collective mechanism R_AB is given by P.sub.{tilde over (X)},{tilde over (Y)}|X,Y=P.sub.{tilde over (X)}|X_P.sub.{tilde over (Y)}|Y where is the Kronecker product. An unbiased estimate of the joint type is given by

T . X , Y = P X ~ Y ~ X , Y - 1 T X ~ , Y ~ = ( ( P X ~ X ^ P X ^ X ) ( P Y ~ Y ^ P Y ^ Y ) ) - 1 T X ~ , Y ~ = ( P X ~ X ^ P X ^ X ) - 1 ( P Y ~ Y ^ P Y ^ Y ) - 1 T X ~ , Y ~ = ( P X ^ X - 1 P Y ^ Y - 1 ) ( P X ~ X ^ - 1 P Y ~ Y ^ - 1 ) T X ~ , Y ~ = ( P X ^ X - 1 P Y ^ Y - 1 ) T . X ^ , Y ^ . ##EQU00010##

[0092] Effect of the Invention

[0093] The embodiments of the invention provide a method for statistically analyzing sanitized private data stored at a server by an authorized, but perhaps, untrusted client in a distributed environment.

[0094] The client can determine empirical joint statistics on distributed databases without compromising the privacy of the data sources. Additionally, a differential privacy guarantee is provided against unauthorized parties accessing the sanitized data.

[0095] Although the invention has been described by way of examples of preferred embodiments, it is to be understood that various other adaptations and modifications can be made within the spirit and scope of the invention. Therefore,

[0096] it is the object of the appended claims to cover all such variations and modifications as come within the true spirit and scope of the invention.

Claims

1. A method for securely determining aggregate statistics on private data, comprising the steps of: randomizing firstly and independently data X and Y to obtain randomized data {circumflex over (X)} and , respectively, wherein the randomizing firstly preserves a privacy of the data X and Y; randomizing secondly independently the randomized data {circumflex over (X)} and to obtain randomized data {tilde over (X)} and {tilde over (Y)} for a server, and helper information T.sub.{tilde over (X)}|{circumflex over (X)} and T.sub.{tilde over (Y)}| [[T.sub. | ]] for a client, respectively, wherein T represents an empirical distribution, and wherein the randomizing secondly preserves the privacy of the aggregate statistics of the data X and Y; determining, at the server, T.sub.{tilde over (X)},{tilde over (Y)}; applying, by the client, the helper information T.sub.{tilde over (X)}|{circumflex over (X)} and T.sub.{tilde over (Y)}| [[T.sub. | ]] to T.sub.{tilde over (X)},{tilde over (Y)} obtain an estimated {dot over (T)}_X,Y, wherein "|" and "," between X and Y represent a conditional and joint distribution, respectively.

2. The method of claim 1, wherein the data X are produced by a first data source, and the data Y are produced by a second data source, and the data X and Y are produced independently in a distributed manner.

3. The method of claim 1, wherein the randomizing uses a Post RAndomisation Method (PRAM).

4. The method of claim 1, wherein the randomizing firstly and secondly are different.

5. The method of claim 1, wherein the helper information is small compared to the data X and Y.

6. The method of claim 1, wherein data X and Y are random sequences, and data pairs (X_i,Y_i) are independently and identically distributed.

7. The method of claim 1, wherein the randomizing preserves differential and distributional privacy of the data X and Y.

8. The method of claim 1, wherein the randomizing secondly provides distributional privacy that is stronger than the differential privacy provided by the randomizing firstly.